# An Introduction to Design and Optimization of Statistical Quality Control

Alternative quality control (QC)
procedures can be applied on a process to test statistically the null hypothesis, that the process conforms to the quality specifications, therefore that the process is in control, against the alternative, that the process is out of control. When a true null hypothesis is rejected, a statistical type I error is committed. We have then a false rejection of a run of the process. The probability of a type I error is called probability of false rejection. When a false null hypothesis is accepted, a statistical type II error is committed. We fail then to detect a significant change in the probability density function of a quality characteristic of the process. The probability of rejection of a false null hypothesis equals the probability of detection of the nonconformity of the process to the quality specifications.

The QC procedure to be designed or optimized can be
formulated as :

*Q**1**(**n**1**,** X**1**) **# Q**2**(**n**2**,** X**2**) **# ... # Q**q**(**n**q**,** X**q**)* (1)

where Qi(ni*,* Xi) denotes a statistical decision rule, ni denotes the
size of the sample Si, that is the number of
the specimens the rule is applied upon, and Xi denotes the vector
of the rule specific parameters, including the decision limits. Each
symbol # denotes either the
Boolean operator AND or the
operator OR. Obviously,
for # denoting
AND, and for n1 <
n2 < ... <
nq, that is for
S1 ⊂
S2 ⊂ ... ⊂ Sq, the (1) denotes a q‑sampling QC procedure.

Each statistical decision rule is evaluated by calculating the respective statistic of the measured quality characteristic of the sample. Then, if the statistic is out of the interval between the decision limits, the decision rule is considered to be true. Many statistics can be used, including the following: a single value, the range, the mean, the standard deviation, the cumulative sum, the smoothed mean, and the smoothed standard deviation. Finally, the QC procedure is evaluated as a Boolean proposition. If it is true, then the null hypothesis is considered to be false, the process is considered to be out of control, and the run is rejected.

A QC procedure is considered to be optimum when it
minimizes (or maximizes) a context specific objective function. The objective
function depends on the probabilities of detection of the nonconformity of the process to the quality specifications and of false rejection. These probabilities depend on the parameters of the QC procedure (1) and on the probability density function of the measured quality characteristic of the process.

The probabilities of detection of the nonconformity of the process to the quality specifications and of false rejection can be:

a. Estimated, using simulation, as it is described in Hatjimihail AT. A tool for the design and evaluation of alternative
quality control procedures. Clin Chem 1992;38:204-10, presenting the first interactive statistical QC simulator in clinical laboratory medicine (see Other publications by HCSL Fellows).

b. Calculated, using numerical methods, as in Hatjimihail AT. Tool for Quality Control Design and Evaluation. Wolfram Demonstrations Project, 2010, presenting a free application available to clinical laboratories (see HCSL publications on statistical QC design).

In general, we can not use algebraic methods to optimize
the QC procedures. Usage of enumerative methods would be very tedious,
especially with multi-rule procedures, as the number of the points of the
parameter space to be searched grows exponentially with the number of the
parameters to be optimized. Optimization methods based on the genetic algorithms (GAs) offer an
appealing alternative as they are robust search algorithms, that do not require
knowledge of the objective function and search through large spaces quickly.
GAs have been derived from the processes of the molecular biology of the gene
and the evolution of life. Their operators, cross-over, mutation, and
reproduction, are isomorphic with the synonymous biological processes. GAs have
been used to solve a variety of complex optimization problems. Furthermore, the
complexity of the design process of novel QC procedures is obviously greater
than the complexity of the optimization of predefined ones. The classifier
systems and the genetic programming paradigm have shown us that GAs
can be used for tasks as complex as the program induction.

In fact, since 1993, we have successfully used the GAs to
optimize and to design novel QC procedures, as it is described in Hatjimihail AT. Genetic algorithms based design and optimization of
statistical quality control procedures. Clin Chem 1993;39:1972-8 (see HCSL publications on the GAs based design and optimization of statistical QC).

Aristeidis T. Chatzimichail,
M.D., Ph.D.,
ath@hcsl.com